The leaked memos, which described the Trump administration as “inept”, led to Sir Kim’s resignation amid the diplomatic furore.
On Thursday, Tory leadership hopeful Mr Johnson said: “I think the real culprit in all this is the person who chose to leak confidential advice given by a great civil servant to the Government and to ministers.
“We need to find whoever did it, hunt them down and prosecute them.”
He added that the leak was “really corrosive of trust in the civil service” .
“I would repeat the point that I made when I talked to him yesterday – I think it is absolutely vital that brilliant ambassadors like Kim are able to speak their minds without fear or favour,” Mr Johnson said.
Mr Johnson’s previously said he thought whoever was behind the leak should be “caught and eviscerated”, after facing some criticism for a lack of support for Sir Kim in a televised debate.
His renewed call comes after the Government and police held “initial discussions” over the investigation into the leak of memos written by the UK’s former ambassador to Washington.
“There have been some initial discussions with the police on the investigation. If there was concern about criminal activity the police would become involved more formally at that point,” a No 10 spokeswoman said, in regards to the Whitehall investigation.
Meanwhile, the inquiry into the leak of diplomatic cables was focusing on whether “someone within the system” was responsible, a Foreign Office minister said on Thursday,
In the Commons, Sir Alan Duncan told MPs: “We do not, at the moment, have any evidence that this was a hack so our focus is on finding someone within the system who has released illicitly these communications… that is where the inquiry is primarily focused at the moment.”
Sir Kim resigned earlier this week and in a letter to the head of the Foreign Office, Sir Simon McDonald, Sir Kim said he felt it was “impossible” for him to carry on in his current role.
President Donald Trump was highly critical of Sir Kim following the leak and said he would no longer deal with him.
He also slammed Prime Minister Theresa May’s Brexit negotiations as he embarked on a twitter tirade following the memos, which were obtained by the Mail on Sunday, being published.